The first step is to enumerate \(2^{O(k^2)}\)  homology classes. Before doing that, they remove several irrelevant vertices from \(G\) 
as a preprocessing step.
Given an instance \((G,T,k)\)  of Planar Disjoint Paths,
this algorithm first removes several irrelevant vertices from \(G\)  so that
the answer of \((G,T,k)\)  remains the same, and the treewidth of \(G\)  decreases to \(2^{O(k)}\) .
Indeed, it is a common ingredient of all previous algorithms for
Planar Disjoint Paths, which is called the irrelevant vertex technique.
More specifically, a vertex \(v\)  of \(G\)  is said to be irrelevant for \((G,T,k)\)  if
\((G,T,k)\)  is a YES-instance if and only if \((G-v, T, k)\)  is a YES-instance,
where \(G-v\)  is the graph obtained from \(G\)  by removing \(v\)  and its incident edges.
Roberson and Seymour [1]} showed
that a planar graph having treewidth \(g(k)\) 
has an irrelevant vertex for a specific function \(g(k)\) . For this, they showed that
a planar graph with treewidth \(O(w)\)  has a \(w\times w\)  grid as a minor.
If \(w\ge g(k)\)  for a specific function \(g(k)\) , there is a grid minor of \(G\)  of size \(g(k)\times g(k)\) 
such that the innermost vertex of the grid minor is irrelevant.
Later, the bound of \(g(k)\)  was improved to \(O(k^{1.5}2^k)\)  by Adler et al. [2]}.
Therefore, by repeatedly removing irrelevant vertices from \(G\) , we can obtain an equivalent instance \((G^{\prime },T,k)\)  where the treewidth of \(G^{\prime }\)  is \(O(g(k))\) .
A single irrelevant vertex can be found in linear time, and thus a naive implementation of this technique yields a quadratic-time algorithm.
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